Make the dough: Place the flour, sugar and yeast in a large bowl. Make a well and pour in the lightly beaten eggs. Mix. Add the very soft butter and mix again until you have a compact dough. Add the rum and form the dough into a ball.
Wrap the dough in a sheet of food-grade plastic, and place it in the refrigerator for 1 hour.
Meanwhile, prepare the cream: split and scrape the vanilla pod to collect the seeds. Place them in a saucepan with the milk and bring to the boil.
In a bowl, whisk the egg yolks with the sugar until the mixture turns white. Add the flour and mix.
As soon as the milk boils, pour it gradually over the mixture, while whisking briskly. Mix until you have a nice thick and homogeneous cream. Add the rum. If the cream is not thick enough, you can put it back on the heat for a few moments, while whisking briskly.
Let the cream cool completely.
Preheat your oven to 180°C. Roll out half of the pastry on your work surface, giving it a thickness of about 5 mm. Place the pastry in a 20 cm diameter cake tin and press the edges of the pastry up against the sides with your fingers.
Pour the cream into the centre of the cake.
Roll out the remaining pastry to a thickness of 5 mm. Cut out a circle of pastry the size of the mould and place it on the cream. Weld the edges between the two pastries so that the cream does not escape.
If you want a golden Basque cake, brush it with egg yolk before baking. Use the back of a fork to make decorative patterns, then bake for 30 minutes.
Leave the Basque cake to cool before eating it with black cherry jam.